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Suppose we are given a fiber bundle $p:E \to B$ and a point $x \in B$. Denote by $p \big|_{p^{-1}(x)}:p^{-1}(x) \to B$ the restricted fiber bundle and by $\Gamma^0(p)$ (resp. $\Gamma^0(p \big|_{p^{-1}(x)})$) the space of continuous sections with the compact-open topology. Is the restriction map $\rho:\Gamma^0(p)\to\Gamma^0(p\big|_{p^{-1}(x)})$ in general a weak homotopy equivalence or not? I'm obviously having trouble with some basic things but any help is well appreciated.

UPDATE: Just to make clear what I have so far: since we're considering the restriction to one point we have that $\Gamma^0(p\big|_{p^{-1}(x)})\cong p^{-1}(x)$ and thus it seems to me highly unlikely that $\rho$ might be a weak homotopy equivalence in general. Except if I'm missing some very basic fact about fiber bundles which is what I'm here to check for.

UPDATE 2: I just tried to prove surjectivity of the induced map $\rho_*$ on homotopy in a general setting but was unsuccessful, hinting at either a wrong assumption or an inept mathematician.

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@willie Thanks for the suggestion, it looks like the software (or more likely some astute moderator) realized that the unregistered Martin and the newly registered Martin are the same. – Martin Worsek Jun 28 '11 at 15:01
that was me. A good samaritan was nice enough to ping me when he noticed that you have both a registered and an unregistered account. – Willie Wong Jun 29 '11 at 9:50
up vote 0 down vote accepted

The statement in the original paragraph is very obviously not true in general. Instead assuming additionally contractibility of the base space one can very easily show that restriction-to-a-fiber/evaluation maps of sections do constitute weak homotopy equivalences. Thanks go to Dan Ramras for pointing that out to me.

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